Fluctuations of $N$-particle quantum dynamics around the nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the fluctuations in the quantum dynamics of N bosons with singular interactions, approximating the many-body evolution using a nonlinear Schrödinger equation and a quadratic Fock space evolution.

Contribution

It introduces a norm-approximation for the N-boson dynamics around the nonlinear Schrödinger equation for a broad class of interaction potentials.

Findings

Provides a rigorous approximation of many-body quantum evolution.

Extends analysis to singular two-body interactions.

Connects Bose-Einstein condensation with fluctuation dynamics.

Abstract

We consider a system of $N$ bosons interacting through a singular two-body potential scaling with $N$ and having the form $N^{3\beta-1} V (N^\beta x)$, for an arbitrary parameter $\beta \in (0,1)$. We provide a norm-approximation for the many-body evolution of initial data exhibiting Bose-Einstein condensation in terms of a cubic nonlinear Schr\"odinger equation for the condensate wave function and of a unitary Fock space evolution with a generator quadratic in creation and annihilation operators for the fluctuations.